Can I export multiple objects with individual animations to FBX?
According to Bastien Montagne (https://developer.blender.org/T65768):


  No bug here, to achieve what you want you need to disable All Actions and NLA Strips options, then you'll get current scene's animation exported properly. Those two options are used to export several animations (usually for a single object), e.g. to export a character and all its animations to a game engine.


So, to export multiple objects with individual animations, do the following:


Click File > Export > FBX (.fbx)
In the Export FBX options, click the Animation button to view animation options
Make sure NLA Strips and All Actions are unchecked